I wanted to clarify a rule that I've seen a lot of people get wrong. When you use a character ability to draw face-up clues from the line, you only replace the cards at the END of your turn.
For example, if you use the ability with Lestrade (pay 3 invest tokens to draw 2 face-up clues), you pick 2 cards from the 4 shown. The line gets replaced after you have chosen your two cards.
The only exception to this is when using the Sherlock/ Mycroft optional card to reserve cards from the line. This card can be used at any time during your turn (before of after moving a meeple). For example, you use Lestrade to draw two cards. The line is then replenished to four cards. At this point, you can reserve a card from the 4 shown, and then the line is replenished once again before going back to your opponents turn.